Note that \(\tilde{B}(h)\)  is the intersection of all sets \(P_{(S,T)}(h)\)  for all orthants \((S,T)\)  of \(\mathbb {R}^N\) . Moreover, each base polyhedron \(B_{(S,T)}(h)\)  is contained in \(\tilde{B}(h)\)  since \(x \in B_{(S,T)}(h)\)  implies for any \((X,Y) \in 3^N\)  that (see also [1]} and Figure REF )
\(x(X) - x(Y) - h(X,Y) &= x(X) - x(Y) - h(X,Y) + x(S) - x(T) - h(S,T) \\& \le x(S \cup X) - x(T \cup Y) - h((X \cup S) \backslash (Y \cup T), (Y \cup T) \backslash (X \cup S)) \\& \quad + x(S \cap X) - (T \cap Y) - h(X \cap S, Y \cap T) \\& = x(S \backslash Y) - x(T \backslash X)- h(X \backslash Y, T \backslash X) \\& \quad + x(S \cap X) - (T \cap Y) - h(X \cap S, Y \cap T) \\&\le 0.\) 
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